Sanna Erika Forsström

Doing Mathematics with Robots:

an Activity Theoretical Perspective on the Links

between Mathematics and Programming in

Classroom Activities

Thesis for the degree Philosophiae Doctor (Ph.D.) at the

Western Norway University of Applied Sciences

Disputation: 21.10.2020

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© copyright (Sanna Erika Forsström)

The material in this report is covered by copyright law.

Year: (2020)

Title: (Doing Mathematics with Robots)

(an Activity Theoretical Perspective on the Links between Mathematics

and Programming in Classroom Activities)

Author: (Sanna Erika Forsström)

Print: Bodoni AS / Western Norway University of Applied Sciences

ISBN 978-82-93677-22-2

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Scientific environment

This thesis is written as a part of studies in PhD-program Bildung and Pedagogical

Practices in Western Norway University of Applied Sciences. The author is

employed by Østfold University College, Faculty of Education.

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Acknowledgements

Just as this thesis concentrates on collective learning processes, this PhD project

also has been a collective process, which I could not have managed alone. I shared

this process with several people whom I want to thank.

First, I want to thank my supervisors. Geir Afdal, you really have all the superpowers

that a supervisor can have. Thank you for your support, clever and wise advice,

valuable comments, engagement, and time. Your positive and constructive feedback

made this project possible. Odd Tore Kaufmann, thank you for always making time

for answering my questions, reading the manuscripts, and giving valuable

comments. Thank you for your wonderful collaboration. Thank you, Grete

Netteland, for your valuable comments and discussions during the writing process.

Thank you all who read my texts during this process. Thank you, Tamsin Meaney,

Rune Herheim, and Roger Säljö, for your valuable contributions by reading and

commenting on my texts.

Thanks to Østfold University College and the PhD program of Bildung and

Pedagogical Practices at the Western Norway University of Applied Sciences for the

opportunity to complete this study. Thank you, Dean Kjersti Berggraf Jacobsen, at

Østfold University College for your support during this process.

Thank you, my colleague, Hege Marie Mandt, for the fellowship, long discussions,

and sharing of joys and frustrations during this process. Many of the steps of this

journey would not have been possible without you. Thank you, Marianne Maugesten

and the whole mathematics education department, at Østfold University College for

your support during this journey. Thank you also, my other colleagues at Østfold

University College. Special thanks to Hilde Wågsås Afdal for your positive and

supportive comments every time we met.

Thank you, my friends in Finland and Norway. Thank you, Monica Boström and

Tuula Skarstein, for encouraging me to take this challenge and move to Norway.

Thank you for the never-ending discussions about schools, research, and life. I also

want to thank the great academic neighborhood in Isebakke Halden. Thank you,

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Marit Eriksen and Roald Tobiassen, Marte Ree Nolet and Ronald Nolet for your

support for our whole family during this process.

Special thanks for my family and my family-in-law in Finland. Thank you, my

mother-in-law, Liisa Forsström, for babysitting when we moved to Norway and I

was traveling. Thank you, my sister-in-law, Birgitta Forsström for all the supportive

talks and your valuable advice on how to use time and energy during the PhD

project. Thank you, my other sister-in-law, Christina Landén, for discussions about

schools, students, and being a teacher.

A great thanks to my dear sister, Marja Lindberg, who went through a PhD project at

the same time with me. It has been a very special and valuable resource to be able to

share thoughts and get support from my own big sister during different phases in

the PhD project.

Thank you, Mom and Dad, Tuula and Tarmo Palmu, for helping me and my family

by babysitting, delighting us with your visits, and always offering a caring home

when we needed a break and wanted to travel to Finland.

To my lovely sons, Lenny and Anselm, and my wonderful and supportive husband,

Peter. Spending time with you brings meaning to life. Everything is possible when I

am just able to spend time with you in the great outdoors. Our hiking, camping,

surfing, sailing, and skiing tours helped me to get my head out from the bubble and

feet on the ground. Thank you for always being there. I love you.

Isebakke, Halden June 10, 2020 Sanna Forsström

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Abstract

This thesis addresses programming integration in mathematics education by

answering the question: what are the links between mathematics and programming

in classroom activities? This thesis consists of three articles: one literature review

and two empirical articles. The three articles are framed by an extended abstract.

The literature review discussing the educational potential of programming in

mathematics education, brings out the need for discussing students’ learning

processes instead of learning results. There is a need for discussion about the

influence of the teacher role and collaboration between students during the students’

learning processes in programming activities. The two empirical articles address the

students’ collective learning processes in mathematics when Lego Mindstorms

robots are introduced in a classroom of students aged 12 to 15, with a mathematics

teacher who is a novice in programming.

The data from this study consists of ethnographic videotaped material of students’

classroom activities with robots gathered in one lower-secondary school in Norway;

field notes from the classroom observations; and, transcribed video material from

the group interview with three students (the key informants). This data was analyzed

with activity system analysis in Engeström’s (1987) Cultural Historical Activity

Theory (CHAT). The data analysis concentrated on students’ activities with robots

on a micro-level. The focus was on the relationship between different components

during the activity development, such as the role of the teacher, collaboration

between students, the object of the activity and the tools.

According to the findings of this study, mathematics can be linked with

programming activities through the active and negotiating role of the teacher. These

findings contribute to the debate in earlier studies regarding the transformational

potential of digital technology in mathematics education. Earlier studies argued that

it is unclear, how the potential links between digital technology and mathematics

can be exploited. The findings contribute to this discussion by suggesting that the

links between mathematics and programming activities have a transformational, yet

not self-evident, potential in mathematics education. The study demonstrated that,

during programming activities, mathematics could become the alive and

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transformative object of the activity. The fruitful activity development took place

when the teacher and the students collaborated.

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Sammendrag

Denne avhandlingen handler om programmering i matematikkundervisning og tar

for seg følgende problemstilling: hva er koblingene mellom matematikk og

programmering i klasseromsaktiviteter? Denne avhandlingen består av tre artikler:

én er en litteraturgjennom, mens de to resterende er empiriske artikler. I tillegg til

artiklene består avhandlingen av et utvidet abstrakt.

Artikkel 1 gir en litteraturgjennomgang rundt det pedagogiske potensialet ved

programmering i matematikkundervisning, samt viser frem behovet for å diskutere

studenters læringsprosesser i stedet for læringsresultater. Det er behov for mer

forskning innen påvirkningen av lærerens rolle, og samarbeid mellom elevene i

deres læringsprosesser innen programmeringsaktiviteter.

De to empiriske artiklene tar for seg studentenes kollektive læringsprosesser i

matematikk da Lego Mindstorm-roboter ble introdusert i et klasserommet for elever

i alderen 12-15 år og hvor matematikklæreren er en nybegynner i programmering.

Dataene fra denne studien består av etnografisk datamateriale på video om elevenes

aktiviteter i klasserommet med roboter på en ungdomsskole i Norge, feltnotater fra

klasseromsobservasjonene, og transkribert videomateriale fra gruppeintervjuet med

tre elever (hovedinformantene). Dataen ble analysert med aktivitetssystemanalyse i

Engeströms (1987) Cultural Historical Activity Theory (CHAT). Dataanalysen

konsentrerer seg på studenters aktiviteter med roboter på mikronivå. Fokuset var på

forholdet mellom ulike komponenter under aktivitetsutviklingen, for eksempel

lærerens rolle, samarbeid mellom elevene, verktøyene og objektet i aktiviteten.

Ifølge funnene kan matematikk knyttes til programmeringsaktiviteter gjennom

lærerens aktive og forhandlende rolle. Disse funnene bidrar til debatt fra tidligere

studier om teknologiens transformasjonspotensial i matematikkundervisningen.

Tidligere studier hevdet at det er uklart hvordan potensielle koblinger mellom

teknologi og matematikk kan utnyttes. Funnene i denne studien bidrar til denne

diskusjonen ved å antyde at koblingene mellom matematikk og

programmeringsaktiviteter har et transformasjonspotensialet i

matematikkundervisning, men at det er ikke selvinnlysende. Det er vist i studien at

under programmerings aktiviteter har matematikk en mulighet til å bli det "levende"

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og transformative objektet i aktiviteten. En fruktbar aktivitetsutvikling fant sted når

læreren og elevene samarbeidet.

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List of publications

Forsström, S. E., & Kaufmann, O. T. (2018): “A Literature Review Exploring the use of Programming in Mathematics Education.” International Journal of Learning, Teaching and Educational Research, 17(12), 18–32. https://doi.org/10.26803/ijlter.17.12.2

Forsström, S. E. & Afdal, G. (2019): "Learning mathematics through activities with robots.” Digital Experiences in Mathematics Education. https://doi.org/10.1007/s40751-019-00057-0

Forsström, S. E. (2019): “Role of teachers in students’ mathematics learning processes upon the integration of robots.” Learning, Culture and Social Interaction, 21, 378-389. https://doi.org/10.1016/j.lcsi.2019.04.005

Reprints were made with permission from Learning, Culture and Social

Interaction and Digital Experiences in Mathematics Education.

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Contents

Scientific environment .................................................................................................. 2

Acknowledgements ....................................................................................................... 3

Abstract ......................................................................................................................... 5

Sammendrag ................................................................................................................. 7

List of publications ........................................................................................................ 9

PART I: Extended abstract .......................................................................................... 15

1. Introduction .............................................................................................................. 16

1.1 Aim and the overall research question ................................................................ 16

1.2 My background as a mathematics teacher ........................................................ 20

1.3 Literature review (Article 1): research on the educational potential of

programming in mathematics education ................................................................ 22

1.4 Research questions, articles, and the research strategy .................................... 23

1.5 Theory and methods .......................................................................................... 27

1.6 The study design ................................................................................................ 30

1.7 Outline of the thesis ............................................................................................ 31

2. Context .................................................................................................................... 33

2.1 Issues in mathematics education ....................................................................... 36

2.2 Digital Technology in Mathematics Education ................................................. 39

2.3 Robots in mathematics education ..................................................................... 42

2.4 Mathematics and digital technology in the Norwegian school ......................... 44

3. Theoretical framework ............................................................................................ 45

3.1 CHAT as a process theory .................................................................................. 45

3.2 The history of CHAT .......................................................................................... 47

3.3 Components in the activity system analysis ..................................................... 48

3.3.1 Subject of the activity .................................................................................. 48

3.3.2 Object of the activity ................................................................................... 49

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3.3.3 Tools in the activity ..................................................................................... 56

3.3.4 Division of labor ...........................................................................................57

3.3.5 Rules ............................................................................................................ 58

3.3.6 Community .................................................................................................. 59

3.3.7 Outcome ...................................................................................................... 59

3.4 The character of CHAT................................................................................... 60

3.4.1 Activity system analysis ............................................................................... 60

3.4.2 Multi-voicedness ......................................................................................... 60

3.4.3 Historicity ..................................................................................................... 61

3.4.4 Contradictions and expansive transformations ......................................... 62

3.5 Theory use in the articles ................................................................................... 63

3.5.1 Article 1 ........................................................................................................ 63

3.5.2 Article 2 ....................................................................................................... 64

3.5.3 Article 3 ....................................................................................................... 64

4. Methodology ........................................................................................................... 66

4.1 Focused ethnographic research strategy ........................................................... 67

4.2 The five principles in CHAT .............................................................................. 70

4.2.1 Activity system analysis ............................................................................... 70

4.2.2 Multi-voicedness .......................................................................................... 71

4.2.3 Historicity .................................................................................................... 72

4.2.4 Contradictions and expansive transformations ......................................... 72

4.3 The data and methods ....................................................................................... 73

4.3.1 The sample of this study .............................................................................. 73

4.3.2 Data collection and my role as a researcher ................................................ 77

4.3.3 Data analysis ............................................................................................... 80

4.4 The quality of my study ...................................................................................... 91

4.4.1 Validity, reliability and generalization of this study ................................... 92

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4.4.2 Ethical reflections ....................................................................................... 96

5. Summary of the articles ........................................................................................ 100

5.1 Article 2: Learning mathematics through activities with robots .................... 100

5.2 Article 3: Role of teachers in students’ mathematics learning processes upon

the integration of robots ......................................................................................... 101

6. Discussion ..............................................................................................................105

6.1. Findings across the articles ..............................................................................105

6.2 The influence of the role of the teacher on the object of the activity.............. 108

6.3 The influence of the role of the teacher on the mathematical tools in use ..... 109

6.4 The main argument ......................................................................................... 109

6.4.1 Why to transform practices in mathematics education .............................. 111

6.4.2 Into what might mathematics teaching and learning be transformed? ... 112

6.4.3 By whom would the transformations be made? ........................................ 113

6.5 Suggestions for future studies .......................................................................... 114

6.6 Possible implications for policy and practice ................................................... 115

6.6.1 Links between mathematics and programming activities ......................... 115

6.6.2 Time ............................................................................................................ 116

6.6.3 The background of the teacher .................................................................. 117

6.6.4 How programming influences students’ learning ..................................... 117

Sources of data ........................................................................................................... 119

Appendix 1. ................................................................................................................. 127

Appendix 2. ............................................................................................................... 128

PART II: Articles ........................................................................................................ 129

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Tables and figures:

Table 1. The articles and research questions of this study ......................................... 26

Table 2. Components in the Activity System Analysis (retrieved from Article 2) ..... 29

Table 3. Differences between a traditional ethnography and a focused ethnography

adapted from Knoblauch (2005, p. 4). ....................................................................... 70

Table 4. The participation types and involvement (Spradley, 1980, p. 58). .............. 80

Table 5. A short presentation of sessions 4 to 7 ......................................................... 84

Table 6. The activity development during sessions 6 and 7 (retrieved from Article 2)

..................................................................................................................................... 89

Table 7. The comparison of sessions 5 and 6 regarding the role of the teacher

(retrieved from Article 3) ............................................................................................. 91

Table 8. Summary of the findings regarding the relationships between the role of the

teacher and object development, the tools in use, and collaboration between

students (retrieved from Article 3) ........................................................................... 103

Figure 1. The students installing and testing the touch sensors ................................. 16

Figure 2. The student using mathematical tools in order to solve their problem ...... 17

Figure 3. Reconstruction of what the students wrote on the whiteboard................... 17

Figure 4. The students succeeding with their task to program the robot to drive a

circle with a radius of one meter..................................................................................18

Figure 5. The students were excited when they successfully solved the problem ......18

Figure 6. The Activity System Model (Engeström, 1987, p.78) ................................. 28

Figure 7. Research design ............................................................................................ 31

Figure 8. The model of traditional mathematics classroom, reconstructed from

Engeström (2008, p. 89). ........................................................................................... 38

Figure 9. Vygotsky’s model of mediated action, adapted from Engeström (2005, p.

60) and Yamagata-Lynch (2010, p. 17) ...................................................................... 47

Figure 10. Two interacting activity systems with boundary object, reconstructed

from Engeström (2005, p. 63). ................................................................................... 55

Figure 11. Maxwells’s (2005, p. 217) Interactive Model of Research Design, modified

for my study ................................................................................................................ 66

Figure 12. Different ethnographical research scopes, retrieved from Spradley (1980,

p. 30) ........................................................................................................................... 68

file:///C:/Users/sannaf/Desktop/kaikki%20kansiot/vaitoskirja/muutokset%20kevat%202020/uusi%20submitointi%20kesakuu%202020/vaitoskirja%20080620.docx%23_Toc42525997file:///C:/Users/sannaf/Desktop/kaikki%20kansiot/vaitoskirja/muutokset%20kevat%202020/uusi%20submitointi%20kesakuu%202020/vaitoskirja%20080620.docx%23_Toc42525997

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Figure 13. Summary of the codes. ............................................................................... 87

Figure 14. The model of successful robot integration from the perspective of the

activity system analysis in CHAT (Engeström, 1987) .............................................. 108

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PART I: Extended abstract

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1. Introduction

1.1 Aim and the overall research question

“It was not allowed to use a mobile phone (in order to steer the robot),” one of the

students answered when asked how the group of students got the idea to use touch

sensors in order to control the robot. The students’ idea was innovative. Instead of

programming each turn individually in different paths, the students wanted to make

one universal program. The idea was that when the touch sensor was connected to

the left-hand side and pressed down the robot would turn left; otherwise, the robot

would drive straight forward. The right-hand side would work similarly. Figure 1

demonstrates the students installing these touch sensors and is an example from the

data of this study where Lego Mindstorms robots, which can be programmed in the

EV3-programming environment, were introduced to the students aged 12 to 15. The

students were tasked to program the robot to drive a particular path; but, instead of

programming the robot, the students were tempted to use an application on their

smartphone to steer it. Based on the students’ earlier knowledge derived from

outside of the classroom, they knew that the robot could be controlled with a

smartphone application. However, the students were not allowed to use the

application given that the idea was to program the robot. Thus, the students got the

idea to make their own “application.”

Figure 1. The students installing and testing the touch sensors

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According to Hoyles (2018), the integration of digital technology in mathematics

classrooms has the potential to bridge classroom mathematics and the students’

world outside of the classroom. During the session described above, the students

indeed connected their activities with the robot to their “smartphone application

world” outside of the classroom. Still, the links between classroom mathematics and

the activities with the robot were not particularly visible during the session

described above. However, the systematical use of mathematical tools in robot-

based activities were visible in another session where students programmed the

robot to drive a circle with a one-meter radius. In order to achieve this, they used

proportions and circle geometry (see Figures 2 and 3). The students succeeded in

their task (see Figure 4) and were very excited about their success (see Figure 5).

Figure 2. The student using mathematical tools in order to solve their problem

Figure 3. Reconstruction of what the students wrote on the whiteboard

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Figure 4. The students succeeding with their task to program the robot to drive a circle with a radius of one meter

Figure 5. The students were excited when they successfully solved the problem

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So, the links between robot-based activities and classroom mathematics were not

self-evident. The links were constituted by many factors, such as the role of the

teacher, collaboration between students and the tools which were used.

Computer programming, or coding, is defined in Balanskat and Engelhardt (2015) as

a process of producing instructions in a programming language for a computer to do

tasks, problem-solving and create interaction. Several countries, such as Finland and

Sweden, have integrated programming in the mathematics curriculum (Bocconi,

Chioccariello, & Earp, 2018). Norway is in the planning phase of doing so

(Utdanningsdirektoratet, 2019). The majority of countries integrate programming in

their curriculum in order to foster what is called students’ 21st-century skills

(Balanskat & Engelhardt, 2015). Desired such future skills, also referred to as 2030-

century skills in the literature, relate not only to technological knowledge but also to

social skills and creativity (Balanskat & Engelhardt, 2015; OECD, 2018).

In addition to this, the national curriculums, for instance in Finland, Sweden and

Norway, call for connections between curriculum mathematics and programming

activities (Opetushallitus, 2014; Skolverket, 2018; Utdanningsdirektoratet, 2019).

However, the integration represents practical, everyday challenges firstly because a

very new curricular activity is assumed to take time and energy from other activities

in the mathematics curriculum. Secondly, because the role of the teacher may be

challenged if the mathematics teacher does not have a relevant programming

background (Bocconi et al., 2018), thirdly, it is unclear how programming can be

linked to different subject areas (Balanskat & Engelhardt, 2015; Bocconi et al., 2018)

in mathematics. Moreover, fourthly, it is also unclear how programming influences

students’ learning (Balanskat & Engelhardt, 2015; Bocconi et al., 2018) in

mathematics.

This study is an empirical contribution to the knowledge of the potential links

between curriculum mathematics and programming activities in lower secondary

classrooms. It discusses practical, everyday situations in the classroom and

considers the possible challenges in programming integration, such as its

connections with the mathematics curriculum and the role of the teacher. This study

aims to answer the overall research question:

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What are the links between mathematics and programming in classroom

activities?

The relevance of the findings of this study will be discussed by addressing issues in a

traditional mathematics classroom. One critical issue in mathematics education is

the difficulty in motivating students to learn formal curriculum mathematics. Hoyles

(2018) pointed out that students often connect curriculum mathematics with

abstract procedures and rules without making connections outside of the classroom.

However, the students’ motivation to learn mathematics derives from the role of

mathematics in the students’ lives outside of the classroom. Students sometimes

have difficulties in understanding the importance of learning mathematics, if the

mathematics they have to learn does not have any concrete role in their lives (Gellert

& Jablonka, 2009; Gura, 2007; Hoyles, 2016).

Computer programming falls under the umbrella of digital technology. According to

Hoyles (2018), integration of digital technology has the potential to transform

learning and teaching activities in mathematics classrooms by providing outside-of-

classroom connections for mathematics. Hoyles (2018) argued that digital

technology as a tool has the potential to enhance the students’ conceptual

engagement in mathematics. However, Drijvers (2018) responded to Hoyles (2018)

by claiming that this is not that self-evident, because “not enough is yet known about

how to exploit this link between mathematics and tool use, and the way in which this

transforms practices for the sake of mathematical learning” (Drijvers, 2018, p. 230).

This study contributes to this discussion by addressing the potential links between

curriculum mathematics and programming activities in the everyday classroom

context.

1.2 My background as a mathematics teacher

My interest in this topic stems from my background as a mathematics teacher. I

have over 10 years of experience in teaching mathematics in Finland and one in

Sweden. This experience was for different schools and school levels. Throughout my

work experience, I have met different kinds of students, become familiar with

different types of learners and seen many different learning environments. Despite

school levels or students, the same challenge remained present: how can I motivate

students to learn mathematics. Conversely, the students themselves also had a

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recurring question: for what purpose do we need this? As a mathematics teacher, to

answer their question, I referred to activities outside of the classroom in order to

motivate students to learn mathematics. However, I found it challenging to find

connections, which are purposeful for students.

Before moving to Norway and embarking on my Ph.D. studies, I was working as a

mathematics teacher in one lower-secondary school in Finland. Coincidentally,

Finland was in the planning phase of integrating programming as a part of the

mathematics curriculum during the last couple of my teaching years. The planning

included teacher courses, and many of the schools were ready to test programming. I

attended an introduction day for the teachers; and, after a short introduction of Lego

Mindstorms robot, I was excited to test them with my students aged 12 to 15. The

students and I tested the robots together in the mathematics classroom for two

years. The approach was very student-centered: students were able to find their

projects and tasks. In turn, as the teacher, I supported them, guided them, and

helped them with programming when needed. These robot-based activities took

place alongside other activities in the classroom and were mostly extracurricular.

Although I saw robot-based activities as an excellent supplement for traditional

classroom activities, I did not see the direct connection with curriculum

mathematics and robot-based activities.

To illustrate this thought process, I use the example of a student I taught whom I

will henceforth refer to as Pekka. Pekka was not motivated by curriculum

mathematics. I introduced Lego Mindstorms robots in his class with the idea that we

would learn to handle them together. Students had time to become familiar with

robots alongside traditional school mathematics. They were able to create their ideas

about what to do with the robots working in teams of two or three students. So,

Pekka and his team decided to build a car and start programming it. Pekka took

charge when they were programming, and after a couple of attempts, they started to

program their car to do pocket parking. They used trial and error to achieve their

goal. After each trial, the students negotiated with each other and shared ideas about

the next steps and how to move forward. There was also a time to negotiate the point

at which the project was deemed successful, or “ready,” given that this could be a

never-ending problem; you can always be a bit better at parking.

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The point I do want to make with the case of Pekka is that even though I had never

seen Pekka as excited and spontaneous with many ideas and thoughts in the

mathematics classroom as he was with this task, I had difficulties, as a mathematics

teacher, to see the links between curriculum mathematics (my primary concern) and

robot-based activities. Even if I, as a mathematics teacher, saw how students found

robot-based activities motivating, I did not see connections between these two

different components in my classroom. Thus, according to my preconceptions based

on my teaching experience, I was curious but at the same time skeptical about the

usefulness of robots and programming in mathematics education.

1.3 Literature review (Article 1): research on the educational

potential of programming in mathematics education

In order to discern what earlier studies have revealed about the potential links

between the mathematics and programming activities, a literature review of research

was conducted, together with Odd Tore Kaufmann, relevant to the following

question:

What is the educational potential of programming in mathematics education?

From the results of our systematic search, we identified and analyzed 15 articles with

different study types, themes, and designs. Based on these, we identified three

dominant themes: (1) the motivation to learn mathematics, (2) student performance

in mathematics, and (3) the collaboration between students and the transforming

role of the teacher. According to the results of these studies, programming

integration improves students’ motivation to learn mathematics and students’

performance in mathematics for some of the groups of students. We concluded that

earlier studies concentrated mostly on individual learning results and motivation.

Although the articles reported that collaboration between students was widely used

in programming activities and that the role of the teacher was different than before

the integration of programming in the classroom, the influence of these components

for students’ learning was not discussed. We argued that there is a need for studies

analyzing students’ collective learning processes, instead of relying on research that

sees individual learning results as sole indicators, in order to get more detailed

information about the potential links between curriculum mathematics and

programming integration. Collaboration between students has been widely used in

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programming activities, and the role of the teacher has also been found to differ

from traditional; therefore, discussion is needed about the effect of these

components in students’ learning processes (Article 1).

Having reviewed the literature, we searched for a more detailed understanding of

the links between curriculum mathematics and programming activities by

concentrating on analyzing students’ collective activities with robots on a micro-level

in order to understand their learning. We viewed learning as something that can be

understood through analyzing collective processes instead of merely attending to

results on paper-and-pencil tests. We concentrated on interactions between

students, teacher, programming tools and robots.

1.4 Research questions, articles, and the research strategy

In order to find out more about the potential links between curriculum mathematics

and programming activities in the classroom, I conducted a study, with the features

of an ethnographic and intervention study, in one lower-secondary classroom in

Norway, given that Norway is planning to integrate programming into their

mathematics curriculum. This study corresponds to the everyday situation where the

mathematics teacher, who does not have a relevant programming background,

integrates programming in their classroom. One way to make programming

integration somewhat smoother when the teacher does not have extensive training

in programming is to use visual programming environments to get started (Bocconi

et al., 2018). Thus, this study concentrates on the integration of Lego Mindstorms

robots in a classroom, where the teacher and the students did not have any extensive

training. As previously mentioned, Lego Mindstorms robots can be programmed in

the EV3-programming environment; functionally, this means that the steering of

robot motors and utilization of different sensors take place through different visual

blocks (Bocconi et al., 2018).

This study has ethnographical features with me as the researcher in the same social

space (classroom) as the students and teacher. More specifically, the research

strategy in this study is called a focused ethnography, with focused observations, key

informants, and more time-intensive fieldwork than in a traditional ethnography

(Skårås, 2018). Also, my role as a researcher differed from what is the case in

traditional ethnography. I introduced Lego Mindstorms robots shortly for the

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mathematics teacher; and thus, this study has also features of an intervention study.

Following this, the teacher introduced robots in their elective study classroom for

students aged 12 to 15.

Moreover, elective study program was chosen because the free environment of a

class activity provided an opportunity for innovative learning processes without any

pressure from the mathematics curriculum. I followed the students working with

robots by videotaping the activities and writing field notes during one semester,

once a week during 75-minute sessions. That was the time needed to gather insights

into the use of mathematics in the classroom. My observations concentrated on the

key informants, one group of three students, Jacob, Lucas, and Oscar, aged 12 to 13,

and their learning activities with robots. The specific focus on one group of students

made it possible to gain an understanding of the learning processes on a micro-level.

The micro-level observations made it possible to analyze students’ activities in detail

by focusing on students’ communication and interactions with different gestures.

This gave valuable information about the links between mathematics and robot-

based activities.

In order to get a more detailed understanding of the potential links between

mathematics and robot-based activities, the overall research question is discussed in

more detail in two additional articles with two separate research questions (see

Table 1). The discussion about mathematics in use and the role of the teacher are

thus divided into two different articles. Even if articles 2 and 3 discuss the same

students and teacher, these articles focus partially on different sessions. Moreover,

the articles focus on different components in robot integration. The focus in Article

2, “Learning Mathematics Through Activities with Robots,” is on mathematical tools

in use and components influencing it, coupled with activity development in students’

learning processes with mathematical tools in use. The focus in Article 3, with the

title “Role of Teachers in Students’ Mathematics Learning Processes upon the

Integration of Robots,” is the role of the teacher in the students’ learning processes.

So, articles 2 and 3 contribute to different discussions.

The links between mathematics and robot-based activities are discussed in Article 2

with the help of the concept of tool in Cultural Historical Activity Theory (CHAT). I

will present CHAT in more detail in the following subsection entitled Theory and

25

Methods, and in third chapter, entitled Theoretical Framework; but, shortly, the

central unit of analysis in CHAT is a tool mediated collective activity system. The

concept of object has a central role in activity system analysis as a motive and

direction for the collective activity (Engeström, 1987; Roth & Radford, 2011). The

use of tools is constituted1 by the object, the drive, or direction of the activity. Thus,

the focus in Article 2 is on the relationship between mathematical tools and object of

the activity. The article answers the question: what is the relationship between

mathematical tools and object in robot-based collective student learning activities

in secondary education?

Article 3 contributes to the discussion about the role of teacher in fruitful integration

of digital technology, and the potential of educational technology to change the

mathematics classroom into a more student-centered one. According to earlier

studies, integration of digital technology has the potential to make mathematics

classrooms more student-centered and to increase the motivation of students.

However, the change depends on the role of the teacher (Bray & Tangney, 2017;

Olive et al., 2010). Moreover, Article 3 discusses the situation where the teacher does

not have extensive training in programming. It answers the question: how does the

role of the teacher in robot-based activities influence students’ learning processes

in mathematics? The focus is also on the object of activity, which is discussed more

in detail in the following subsection. As Article 2 discusses the relationship between

tools and object, Article 3 discusses the influence of the role of the teacher on the

objects of students’ activities with the robots.

1 I use the concept of constitute to describe relationships that are not causal but mediating.

26

Table 1. The articles and research questions of this study

Article 1:

Literature Review

Article 2: The Use of Mathematical Tools

Article 3: The Role of the

Teacher

Title A Literature Review Exploring the Use of Programming in Mathematics Education.

Learning Mathematics Through Activities with Robots.

Role of Teachers in Students' Mathematics Learning Processes Based on Robotics Integration.

Research question

What is the educational potential of programming in mathematics education?

What is the relationship between mathematical tools and object in robot-based collective student learning activities in secondary education?

How does the role of the teacher in robot-based activities influence students’ learning processes in mathematics?

Topics Motivation to learn mathematics; students' performance in mathematics; the collaboration between students; the role of the teacher; curriculum connections with mathematics.

The use of mathematical tools; the object of the activity; connections with mathematics curriculum.

The role of the teacher; the collaboration between students; object of the activity; mathematical and technological tools.

Contributes to the discussion about

The article focuses on the potential benefits of programming in mathematics education and unsolved questions regarding them.

The article focuses on the connections between robot-based activities and mathematics curriculum, along with collective learning processes in mathematics through activities with robots.

The article focuses on the role of the teacher in a fruitful robot integration, and the role of the teacher in not a teacher-led classroom when the teacher does not have any programming background.

27

1.5 Theory and methods

In order to understand collective learning processes, this study is based on the socio-

cultural paradigm, which means that knowledge creation is viewed as a social

process with different tools in use (Vygotsky, 1978). Based on the socio-cultural

perspective, the focus of this study is on how students and the teacher use different

kinds of tools. According to the socio-cultural perspective, knowledge, and skills

stem from patterns and insights created over time in societies (Säljö, 2014). Thus,

knowledge creation processes emerge through social interactions, which enable

involvement in historically accumulated cultural patterns and tools, such as different

kinds of psychological, linguistic or physical tools (Säljö, 2014). This means that

learning in this study is seen as a social process involving the use of different kinds

of tools. Knowledge creation processes relate to social interactions and

argumentation.

More specifically, the knowledge creation processes in this study are discussed with

Engeström’s (1987) Cultural Historical Activity Theory (CHAT), where the

knowledge creation processes consist of social, multi-voiced interactions. CHAT is

useful for this study because the focus is on the collective classroom activities

instead of on individual actions. The analysis of collective activities gives the

possibility to get information about relational processes in the classroom, which is

difficult with a more individual starting point. CHAT makes it possible to analyze the

role of mathematics and digital technology as part of classroom activities instead of

just an external object or tool. Furthermore, learning processes in the classroom are

discussed without analytical distinction between teacher and students; CHAT sees

teaching and learning as dialectically intertwined processes (Engeström & Sannino,

2012). The teacher is a part of students’ learning processes, which gives information

about teacher-student relationships in the classroom. Thus, CHAT suits well with

analyzing collective learning processes in the context of robot integration. At the

beginning of the programming integration, frequently both students and the teacher

are novices in programming; and so, learning activities are often collective. In that

kind of situation, the teacher may also take the role of a learner, if they have

insufficient knowledge in programming.

In order to analyze and understand the learning processes described above in detail,

I used the activity system model (see Figure 5) in Engeström’s (1987) CHAT, where

28

the seven components (see Table 2) describing the collective activity are in a

relationship with each other through mediation (Engeström, 2005). For instance, in

the uppermost sub-triangle, tool mediates the subject’s activity towards object. With

the help of tools, subjects are in interaction with the object of the activity

(Engeström, 2008). The social components, rules, community and division of labor

influence collective activities. These components make it possible to address

collective achievements (Engeström, 2008).

Figure 6. The Activity System Model (Engeström, 1987, p.78)

29

Table 2. Components in the Activity System Analysis (retrieved from Article 2)

Component Definition/meaning Examples from this study

Subject The individual or group of people who are engaging in the activity (Yamagata-Lynch, 2010)

The students and the teacher

Object The driving force of the activity (motive and goal) (Engeström, 1987)

Fulfill a task with the robot

Tool The instrument that mediates the activity (Engeström, 1987)

The robot, computer, and mathematical tools

Rules The regulations that are relevant to the activity (Yamagata-Lynch, 2010)

Task assignment, the rules of the mathematics classroom

Community The social group to which the subject belongs to during the activity (Yamagata-Lynch, 2010)

The whole class of students and the teacher (or teachers)

Division of labor How the tasks are shared during the activity (Yamagata-Lynch, 2010)

Collaboration between students, the mediation of the teacher

Outcome The result of the activity (Yamagata-Lynch, 2010)

The robot drives a track as it is programmed

Because this study concentrates on the role of the teacher in students’ learning

processes and links between mathematical tools and the activities with robots, the

focus is specifically on the components of tools, object, and division of labor. The

use of mathematics and other tools is constituted by the object of the activity, which

is the foundation for the whole activity (Engeström, 2005). The role of the teacher

can be discussed through division of labor in the activity system analysis.

30

1.6 The study design

According to Maxwell (2005), the design of a qualitative study consists of

interactions between its goal, research questions, theoretical framework,

methodology, and validity. The design of this qualitative study with connections

between research questions, theoretical framework and a research method are

summarized in Figure 7. This study can be classified under the sociocultural

paradigm, where the theoretical framework of this study, CHAT, belongs also. In the

sociocultural paradigm, knowledge creation is seen as a social process with cultural

mediation, i.e. with cultural tools in use (Säljö, 2014).

Furthermore, the overall research question is discussed with the help of the

components from the activity system analysis in CHAT. The most central concepts

are learning processes, tools, object, the role of the teacher, and collaboration. The

need for discussion about these concepts stems from the literature review article,

which emphasized the need to discuss collective learning processes in mathematics

and the role of the teacher in them, instead of focusing solely on individual learning

outcomes. The focus is on the relationships between the teacher, the tools, the object

of the activity and division of labor in students’ collective learning processes. Article

1 discusses students’ learning and motivation to learn mathematics. Articles 2 and 3

both discuss students’ collective learning processes. In Article 2, the focus is on

mathematical tools in use and students’ collective learning in mathematics. Article 3

focuses on the role of the teacher in students’ collective learning processes. The most

central theoretical concepts in articles 2 and 3 are the concepts of tools and object.

Figure 7 shows how these different themes and components, mathematics learning,

motivation to learn mathematics, collaboration between students and the teacher,

the roles of the teacher are presented in different articles in this study. Figure 7 also

shows how different articles in this study are connected through these themes and

components.

Articles 2 and 3 fit in the sociocultural paradigm. The focused ethnographical data

gathered in order to answer the research questions in articles 2 and 3 have been

analyzed with the help of CHAT.

As the design of this study is a complex interactive model, where each part depends

on each other, I will discuss the connections with other components in my study

31

design later in this thesis. In the section of a theoretical framework, I will reflect in

more detail on the usefulness of CHAT in this study. In the methodology section, I

will present a more detailed research strategy, a focused ethnography. I will present

my data and methods more detailed and reflect, how CHAT and focused

ethnography influenced my data collection and analysis methods.

Figure 7. Research design

1.7 Outline of the thesis

This thesis consists of two parts. The first part clarifies my overall research question,

the aim and the context of the study. Here, the theoretical framework and

methodological approach, as well as the contributions of this study, are discussed in

detail. The second part consists of three articles: one literature review article and

two empirical articles.

In the first part of this thesis, the second chapter, I discuss in which different

contexts I place this study. In the third chapter, I introduce the theoretical

framework used in this study. I discuss how different components and characterized

32

features of CHAT suit this study. In the fourth chapter, the methodological parts of

this study will be presented. The fieldwork based on focused ethnography and my

role as a researcher will be presented in detail, the methods of analysis, and the

reliability and validity of this study will be discussed. The fifth chapter is a summary

of the two empirical articles. In the last chapter, I provide a discussion about the

overall contribution of this thesis towards the field. The overall research question

will be discussed and answered with the help of the contributions of all three

articles.

The second part of this thesis consists of the three articles as separate appendices.

These three different articles contribute to the discussion on the links between

mathematics and programming activities independently from three different

viewpoints.

33

2. Context

In this chapter, I will introduce the context in which this study is situated. As

discussed in the Introduction, this study focuses on collective classroom activities,

and the knowledge-creation processes herein are discussed through the lens of

Engeström’s (1987) Cultural Historical Activity Theory (CHAT). Thus, I will describe

the context of this study by first discussing, how context is understood in CHAT.

A context is often understood to be a predefined and stable frame of a study that is

“out there,” but CHAT understands contextualization as being a dynamic process. As

Nardi (1996, p. 38) stated:

Context is constituted through the enactment of an activity involving people

and artifacts. Context is not an outer container or shell inside of which people

behave in certain ways. People consciously and deliberately generate contexts

(activities) in part through their own objects; hence context is not just “out

there.”

As a theoretical framework of this study, CHAT is discussed in greater detail in the

following chapter, so without any deeper explanation here, I will only mention that

CHAT is characterized by a procedural and a relational perspective (Nardi, 1996),

which means that “context” in CHAT is understood more as a process than as a given

product. In CHAT, context forms in the different relationships within an activity. In

this study, the context is formed in classroom activities, which extend far with time

and space and are in relationship with other activities (Nardi, 1996; Van Oers, 1998);

these activities can still be framed by certain relationships.

Again, Nardi (1996, p. 38) wrote:

A context cannot be reduced to an enumeration of people and artifacts; rather

the specific transformative relationship between people and artifacts,

embodied in the activity theory notion of functional organ, is at the heart of

any definition of context, or activity.

As the focus in this study is on collective classroom activities, the context of this

study is formed in different transformational relationships in these activities and in

transformative relationships between people and tools (i.e., artifacts) in these

34

activities. Even if the context forms within activities, those activities, the

participants, and the tools carry their own histories based on other contexts. Those

classroom activities are in relationship with other activities, both inside and outside

of the classroom, which again, can be connected with many different contexts, such

as curriculum, tests and grades, society, political decisions, and media. In order to

understand collective classroom activities and the transformations and relationships

within the activities and between different activities, the activities need to be

connected with different kinds of contexts inside and outside of the classroom.

Thus, the activities inside the classroom need to be discussed in a wider context, and

this study needs to also be connected with contexts other than just classroom

activity. On the other hand, as the study needs to be located in time and space, all

possible connections or relationships cannot be analyzed and discussed (H. Afdal &

Afdal, 2010). H. Afdal and Afdal (2010) pointed out that the context in educational

research is constituted by the research design and has four actors: the studied object,

the researcher, theory, and the research participants. These actors frame the study

with time, space, and relations and thus make the study manageable.

When contextualizing this study, I, as a researcher, created the context of this study

in a dialogic process with the four actors: the classroom activities as the studied

object, the research questions formulated by me as a researcher, CHAT as a theory,

and the teacher, together with the students, as research participants. In this inner-

dialogic process with me as a researcher, my preconceptions and interests were

challenged by earlier studies and the data material of this study. Thus, based on the

dialogic process with the research questions and observed classroom activities, I

made choices regarding connections and relationships addressed in this study. I

based my decisions on the research questions, the research field, earlier studies, and

practical solutions regarding study design. These frameworks make the study

manageable and conductible.

Based on the dialogic process with research questions and classroom activities, the

contexts established further in this chapter and in this study are as follows: 1) The

research field of and critical issues in mathematics education; 2) Digital technology

in mathematics education; 3) Robots in mathematics education; and 4) Mathematics

35

and digital technology in Norwegian schools. I will justify the choice of these

contexts henceforth.

First, in order to justify Contexts 1 and 2: Based on the research questions, I was out

to find a more detailed understanding of the potential links between mathematics

and programming activities. At the activity level, I was interested in learning more

about the role of mathematics and the role of programming in classroom activities.

The discussion about the role of the mathematics in classroom activities is placed in

the research field of mathematics education. As the focus is also on the role of

programming in classroom activities and potential links between mathematics and

programming activities, the discussions in this study focus on the research field of

digital technology in mathematics education (i.e., Context 2). Furthermore, as I

argued in the Introduction, it is unclear how the potential links between

mathematics and programming activities transform activities in the classroom and

influence students’ learning processes in mathematics; hence, this study focuses on

transformative learning processes with digital technology in the classroom. The

discussions concentrate on the relationships between different components, such as

the role of the teacher, collaboration between students, the role of

programming/robots and the role of mathematics in the collective classroom

activities and the activity development during these activities. In order to discuss the

relevance of the findings of the discussions about how the role of mathematics and

that of the teacher in students’ programming activities are connected to the research

field of digital technology in mathematics education by also discussing critical issues

in mathematics education (i.e., Context 1). As discussed in the Introduction, the

earlier studies have connected the integration of digital technology with critical

issues in mathematics education and discussions on how the integration of digital

technology can contribute through transformative classroom activities to reduce

these issues.

Second, the Lego Mindstorm robots present one way to connect programming in

classroom activities (i.e., Context 3). Robots also bring a more concrete dimension to

programming and Lego Mindstorm robots are the most studied educational robots

(Benitti & Spolaôr, 2017).

36

Third, programming is a current topic in mathematics education in many countries,

especially in Norway at the moment. After the curriculum reform in Norway,

programming is going to be a part of students’ and teachers’ activities in

mathematics classrooms. Thus, the data collection of this study took place in a

classroom in a Norwegian school context (i.e., Context 4) with students and the

teacher as research participants and their activities with robots as a unit of analysis.

The programming topic could be approached from several different angles and

research environments, such as by discussing curriculums, students outside of

classroom activities or students’ improvements in mathematics. However, such

approaches would require a different kind of study design with different kinds of

practical solutions. I found it difficult, for instance, to follow students outside of the

classroom activities, because I did not know the informants of this study beforehand.

The framing of the study with time and space would also prove to be challenging.

The programming activities in a Norwegian classroom environment provide the

possibility to discuss what kind of learning processes are activated when

programming as a new element comes into play.

Thus, different contexts discussed here are issues in mathematics education, digital

technology in mathematics education, and robots in mathematics education. The

Norwegian school context is also presented.

2.1 Issues in mathematics education

Mathematics is an integral part of our lives and our society; mathematics is

everywhere. In research, for instance, it is not only needed when modeling in the

sciences, the social sciences, and in economics, but also many areas of the

humanities. In our everyday lives, on the other hand, we experience it through its

role in tools, such as computers and smartphones. However, students still have

difficulties understanding the importance of learning mathematics, even if

mathematics education is under development, and even though mathematics is

present in digital technology and society (Gellert & Jablonka, 2009; Gura, 2007;

Hoyles, 2016). Even though mathematics is everywhere, mathematics is mostly

invisible in our society (Hoyles, 2015). Several researchers argue that, even though

mathematics is everywhere, mathematics education is not connected with everyday

activities outside of school or other school subjects in many classrooms (Boaler,

37

2009; Bray & Tangney, 2017; Hoyles, 2016). In this context, mathematics

classrooms are often quite teacher-centered, and teachers give students ready-made

tasks to solve (Engeström, 2008; Olive et al., 2010; Valoyes-Chávez, 2019).

An additional issue in a traditional teacher-led mathematics classroom is the

complexities of teaching and learning practices (Hoyles, 2015). According to reform

suggestions in mathematics education, an effective approach in the classroom is

student-centric, and teaching and learning activities are interdependent processes

(Franke, Carpenter, Levi, & Fennema, 2001; Valoyes-Chávez, 2019). This kind of

approach is called reform mathematics, which utilizes a problem-oriented and

inquiry-based approach (Goos, 2004), wherein the teacher meets students’ needs

based on their own interests and previous knowledge (Valoyes-Chávez, 2019). In

reform-oriented, inquiry-based classrooms, the aim is to get students to participate

and engage in the communication, reasoning and problem-solving activities inside

the classroom (Goos, 2004). However, it is not self-evident, “what kinds of practices

do we wish students to participate; and … what specific actions … a teacher [should]

take to improve students’ participation” (Goos, 2004, pp. 281–282). Furthermore,

an additional issue is how the teacher is able to connect activities in the classroom

with students’ lives and cultures outside of the classroom (Goos, 2004; Parker,

Bartell, & Novak, 2017).

Despite curriculum reforms and pedagogical discussions about practical problem-

solving activities and integration of digital technology into school mathematics

(Albert & Kim, 2013; Contreras, 2014), a traditional teacher-led approach connected

with abstract mathematics still remains to be established in an everyday context in

many mathematics classrooms (Albert & Kim, 2013; Bray & Tangney, 2017; Opheim

& Simensen, 2017). A broad range of research argues that students connect the

subject of mathematics with abstract procedures, rules, and memorization, all of

which have a central role in the mathematics classrooms (Albert & Kim, 2013; Bray

& Tangney, 2017; Hoyles, 2016, 2018; Opheim & Simensen, 2017; Pietsch, 2009). In

this context, students often have a passive relationship with their learning, they

follow fixed rules and get fixed, unquestioned answers with fixed numbers (Boaler,

2009; Bray & Tangney, 2017; Ernest, 1996; Hoyles, 2016).

38

This kind of traditional classroom environment, which is usual in mathematics

education, can be illustrated with Engeström’s (1987) activity system model. In a

traditional classroom, the teacher and students have different activity system models

(see Figure 8). In a student’s activity system model, the object of the activity is

traditionally the task given by a teacher, often from the book. The desired common

outcome in both activity systems is test results and grades. This is the only shared

component in the teacher’s and the student’s activity system models. (Engeström,

2008). Engeström (1987) calls objects, which are reproduced in order to gain test

results or grades, dead objects (Engeström, 2008).

To interpret the description above concerning the situation where mathematics is

connected with procedures and rules with Engeström’s (2008) traditional classroom

model (see Figure 8), mathematics has a role as rules or pre-defined tools in a

student’s activity system model. Students are assumed to use specific mathematical

tools with certain rules in order to solve tasks given by the teacher.

Engeström (2008) argues that when discussing the potential to change classroom

culture, the focus should be on the object of education. According to Engeström

(2008), the potential changes in the classroom activities occur through object

constructions, which are influenced by all components in the activity system model.

Thus, it is interesting to uncover whether or not the integration of robotics in

mathematics classrooms has the potential to make changes to the objects of

activities; and if so, how? From a broader perspective, according to earlier studies,

Figure 8. The model of traditional mathematics classroom, reconstructed from Engeström (2008, p. 89).

39

integration of digital technology, in general, has the potential to change a traditional

classroom culture in mathematics education. This potential will be discussed in the

following section.

2.2 Digital Technology in Mathematics Education

Digital technology, such as computers, tablets with different applications, and digital

whiteboards, have been a long-standing staple of mathematics education. This idea

to integrate programming in mathematics classrooms is not new. As Papert

suggested in 1980, programming should be included in the school curriculum. He

argued that mathematics teaching and learning often happens through primitive

technological tools, such as paper and pencil, for instance, used in drawing different

graphs or solving equations with a set of guiding principles. This kind of teaching or

learning practices is not connected to the students’ world outside of the classroom.

Papert’s (1980) idea was to reconstruct learning and teaching processes in

mathematics through programming integration by providing new relationships and

connections with formal mathematical knowledge. According to his suggestions,

mathematical activities through computer programming could transform the roles

in a traditional teacher-led classroom. As he stated it:

Mathematics is a real activity that can be shared by novices and experts. The

activity is so varied, so discovery-rich, that even on the first day of

programming, the student may do something that is new and exciting to the

teacher. (Papert, 1980, p. 179)

After Papert’s suggestion, a lot has happened in mathematics classrooms with

developing digital technology in the classroom. Still, as discussed in the previous

chapter, the traditional classroom model is usual in many mathematics classrooms.

The discussion about the usefulness of digital technology in mathematics education

is still ongoing, and it has taken until now to integrate programming into the school

curriculum of many countries.

According to earlier studies, the integration of digital technology has a potential to

change the traditional teaching and learning processes in mathematics education by

offering connections outside of the classroom and possibility for students to get

ownership of their learning (Bray & Tangney, 2017; Gellert & Jablonka, 2009;

Hoyles, 2016, 2018; Olive et al., 2010). However, that is not self-evident. The links

40

between curriculum mathematics and technology-based activities in the classroom

are not always that simple (Drijvers, 2018). Furthermore, a range of earlier studies

argued that how we integrate digital technology in the mathematics classroom is

critically determined by the teacher’s complex role in the classroom and their

knowledge (Drijvers, 2012, 2015; Drijvers et al., 2014; Ruthven, 2014).

A number of earlier studies approach the teacher’s role and knowledge during

integration of digital technology with the theoretical concept of instrumental

orchestration (Drijvers, 2012; Drijvers, Doorman, Boon, Reed, & Gravemeijer, 2010;

Drijvers et al., 2014; Ruthven, 2014) developed by Trouche (2004) and/or the

TPACK (Technological Pedagogical Content Knowledge) model (Drijvers et al., 2014;

Ruthven, 2014) developed by Mishra and Koehler (2006). These approaches focus

on teacher’s expertise in the classroom (Ruthven, 2014). As the studies using the

concept of instrumental orchestration focus on the teachers’ classroom practices

with digital technology, the TPACK model is used to discuss the teachers’ skills

(Drijvers et al., 2014).

Trouch (2004) defined instrumental orchestration as the teacher’s organization and

management of the use of technological artifacts (i.e., tools) in the classroom in

order to steer students’ instrumental genesis, which is a process wherein artifacts

are distinguished from instruments; artifacts are something that has been given, and

an artifact can become an instrument when a subject applies the artifact in her

activities. Trouche (2004, p. 285) defined the concept of instrument and

construction thereof as follows: “An instrument can be considered as an extension of

the body, a functional organ made up of an artifact component (an artifact, or the

part of an artifact mobilized in the activity) and a psychological component. The

construction of this organ, named instrumental genesis, is a complex process,

needing time, and linked to the artifact characteristics (its potentialities and its

constraints) and to the subject’s activity, his/her knowledge and former method of

working.”

In the instrumental genesis, “tool and person co-evolve so that what starts as a crude

‘artefact’ becomes a functional ‘instrument’ and the person who starts as a naive

operator becomes a proficient user” (Ruthven, 2014, p. 7).

41

According to the studies using the concept of instrumental orchestration, integrating

digital technology into the mathematics classroom may transform teaching

practices, but it is hardly a revolution (Drijvers et al., 2010). When discussing the

different kinds of instrumental orchestrations used by a teacher in a technology-rich

classroom, Drijvers et al. (2010) divided teachers’ different orchestrations into three

different categories: a didactic configuration, an exploitation mode, and a

didactical performance. A didactic configuration handles the ways in which

teaching is didactically organized in the classroom; for instance, which artifacts

(tools) are planned to be used in education. An exploitation mode handles, for

instance, decisions about different tasks or about how artifacts (tools) are intended

to be used. A didactic performance handles decisions made by a teacher in a

teaching situation in the classroom. Drijvers et al. (2010) argued that when new

digital technologies are introduced in the classroom, the teachers’ new practices and

didactic decisions are related to the practices with which they are familiar: the

teachers make their own choices which are related to their regular habits. The

teacher-led approach with closed questions is still more or less present in teaching

practices with digital technology (Drijvers et al., 2014).

Moreover, according to Drijvers et al. (2014), the teachers’ skills are an essential

factor in the successful integration of digital technology. Drijvers et al. (2014) argued

that teachers use their pedagogical-content knowledge alongside their technological

skills, which enables satisfactory integration of this technology in most of the cases.

Still, the teachers who are less experienced and skeptical of the use of digital

technology are an essential group to consider when discussing the potentials of

technology integration. I think this is an interesting point to consider when

discussing programming integration in mathematics education, when mathematics

teachers do not necessarily have any extensive training in programming.

The role of the teacher may also become challenging when linking digital

technology-based activities with curriculum mathematics. Earlier studies argued

that the students’ focus is often on the technological and practical side, while

mathematics teachers’ concerns lie mostly on mathematics (Drijvers et al., 2010;

Lagrange & Ozdemir Erdogan, 2009). When students need much technological

advice from the teacher, the focus on teaching practices is mostly on the digital

technology itself, rather than on mathematics (Drijvers et al., 2010). Drijvers et al.

42

(2014) found that after the resolution of technological issues, there is a place for

mathematics, which creates pressure on the teacher to develop sound technological

skills.

In order to summarize the challenges regarding the transformative potentials of

digital technology integration in mathematics education, I interpret Drijvers’ (2018)

response to Hoyles’ (2018) optimistic ideas about the potentials of digital

technology. Drijvers (2018) claimed that it is not clear enough how the potential

links between mathematics and digital technology-based activities can be exploited

to transform processes in the mathematics classroom. His first concern was the why

question. He claimed that if the reason to transform learning and teaching practices

in mathematics classrooms is to foster students’ measurable outcomes, there is not

enough research-based evidence to do so (Drijvers (2018).

Drijvers’ (2018) second concern was the into what question. He claimed that it is

unclear into what teaching and learning practices should be transformed. Hoyles

(2018) argued that digital technology could provide new windows in mathematics

learning when students are using mathematics in new situations. However, as

discussed earlier in this section, the mathematics behind the digital technology is

not the students’ first concern when they are using the technology. The teacher is

more concentrated on mathematics than students. Thus, the links between digital

technology-based activities and mathematics are still unclear.

Drijvers’ (2018) third concern regarding the transformative power of digital

technology is by whom. Hoyles (2018) suggested that technological tools can

transform classroom practices. However, Drijvers (2018) argued that tools alone

could not make any changes; the potential of digital technology is determined by the

manner in which it is used or planned-to-be-used by teachers or educational

designers. As discussed earlier in this section, teachers’ practices and skills influence

in which manner digital technology will be integrated.

2.3 Robots in mathematics education

There are many different robots or toolkits which can be used educationally (Karim,

Lemaignan, & Mondada, 2015), and Lego Mindstorms robots are the most studied

(Benitti & Spolaôr, 2017).

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This study concentrates on Lego Mindstorms robots because they provide a smooth

way of integrating programming in the earlier stages of integration. The visual

programming environment, where different figures represent different

programming structures (such as loops and if-structure) make the teaching of

programming easier for teachers, who do not have any extensive training in

programming (Bocconi et al., 2018). In the EV3-programming environment, it is

possible to program Lego Mindstorms robots by changing the values of different

variables in different figures (Bocconi et al., 2018).

Educational robotics provides an opportunity outside of the classroom connections

in mathematics education (Ardito, Mosley, & Scollins, 2014; Barak & Assal, 2018);

but, as discussed, the potential of digital technology in mathematics education is not

self-evident. Moreover, the educational benefits and curriculum connections with

robotics remain unclear (Alimisis, 2013; Benitti & Spolaôr, 2017; Savard &

Highfield, 2015). As discussed in our review discussing the educational potential in

mathematics education, the positive effect of programming cannot be generalized, at

least regarding improvement in the students’ achievement in mathematics. The

same trend is visible in studies only discussing robotics and mathematics education.

For instance according to Lindh and Holgersson (2007) results regarding students’

improvement in mathematics after training with Lego Mindstorms robots differed

for different groups of students.

Furthermore, systematic use of formal mathematics is challenging with robots.

Barak and Assal (2018) argue that the learning of formal mathematics and other

STEM subjects can be challenging while using robots. According to Savard and

Freiman (2016), even if students used mathematics in their robot-based activities,

the use of mathematics was not systematic, as students did not design the use of

mathematics in their problem-solving activities with robots. Instead, they started

their problem solving with digital context, using a trial-and-error strategy above a

systematic mathematical approach, which according to Savard and Freiman (2016),

acted as an obstacle for greater mathematical understanding. The trial and error

strategy is defined here as an iterative process where students test and repair the

program without making any detailed plan until they succeed.

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In summary, even if robot integration could provide out of school connections in a

mathematics classroom and enhance comprehension and motivation, the links

between mathematics education and robot-based activities remain unclear.

2.4 Mathematics and digital technology in the Norwegian school

The compulsory school system in Norway consists of a 10-year elementary school.

Students begin their compulsory school in the year they turn six. The subject of

mathematics has a central role in the national curriculum in Norway, where

mathematics is seen as one of the main subjects, is part of cultural heritage and is

seen as the basis of logical thinking. In the curriculum of mathematics, problem-

solving is highlighted, and the use of digital technology is strongly present

(Utdanningsdirektoratet, 2013).

The Norwegian school represents a suitable context for this study because Norway is

planning to integrate programming within their mathematics curriculum

(Utdanningsdirektoratet, 2018). Furthermore, the school system in Norway has a

positive attitude towards the use of digital technology. In addition to the regular use

of digital technology in regular education, students can choose technology as an

elective subject (Utdanningsdirektoratet, 2013). Also, according to the survey about

the use of ICT in education in Europe, Norwegian schools are highly digitally

equipped (Wastiau et al., 2013).

In connection to the suggestion of curriculum reform in 2020 in Norway,

programming is construed as a promising element of mathematics education:

“Through programming, students can be more creative in approaching issues and

gain the ability to explore connections that have not been possible to explore before.”

(Utdanningsdirektoratet, 2019). The Norwegian school context is interesting for this

study also due to the role of the teacher in programming integration. In Norway,

4,154 lower secondary teachers answered the 2018 TALIS (Teaching and Learning

International Survey) conducted by the OECD (2019) in 48 different countries. The

Norwegian teachers’ responses indicated a need for increased technological training

for the teachers (Throndsen, Carlsten, & Björnsson, 2019). It is not enough to give

the teachers new equipment for their classrooms; they also need advice as to how to

appropriately integrate the digital technology in their teaching (Throndsen et al.,

2019).

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3. Theoretical framework

This study concentrates on a detailed analysis of the learning processes. As many of

the earlier studies discuss the role of the teacher regarding integration of digital

technology in mathematics classroom with the concept of instrumental

orchestration or TPACK model, this study takes a broader and more detailed

approach by discussing the role of the teacher in relation to other components in the

classroom.

Ruthven (2014